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Theorem ltrncnvatb 30949
Description: The converse of the lattice translation of an atom is an atom. (Contributed by NM, 2-Jun-2012.)
Hypotheses
Ref Expression
ltrnatb.b  |-  B  =  ( Base `  K
)
ltrnatb.a  |-  A  =  ( Atoms `  K )
ltrnatb.h  |-  H  =  ( LHyp `  K
)
ltrnatb.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
ltrncnvatb  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( P  e.  A  <->  ( `' F `  P )  e.  A
) )

Proof of Theorem ltrncnvatb
StepHypRef Expression
1 ltrnatb.b . . . . . 6  |-  B  =  ( Base `  K
)
2 ltrnatb.h . . . . . 6  |-  H  =  ( LHyp `  K
)
3 ltrnatb.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
41, 2, 3ltrn1o 30935 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F : B
-1-1-onto-> B )
543adant3 975 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  F : B
-1-1-onto-> B )
6 simp3 957 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  P  e.  B )
7 f1ocnvdm 5812 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  P  e.  B )  ->  ( `' F `  P )  e.  B
)
85, 6, 7syl2anc 642 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( `' F `  P )  e.  B )
9 ltrnatb.a . . . 4  |-  A  =  ( Atoms `  K )
101, 9, 2, 3ltrnatb 30948 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( `' F `  P )  e.  B
)  ->  ( ( `' F `  P )  e.  A  <->  ( F `  ( `' F `  P ) )  e.  A ) )
118, 10syld3an3 1227 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( ( `' F `  P )  e.  A  <->  ( F `  ( `' F `  P ) )  e.  A ) )
12 f1ocnvfv2 5809 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  P  e.  B )  ->  ( F `  ( `' F `  P ) )  =  P )
135, 6, 12syl2anc 642 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( F `  ( `' F `  P ) )  =  P )
1413eleq1d 2362 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( ( F `  ( `' F `  P )
)  e.  A  <->  P  e.  A ) )
1511, 14bitr2d 245 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( P  e.  A  <->  ( `' F `  P )  e.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   `'ccnv 4704   -1-1-onto->wf1o 5270   ` cfv 5271   Basecbs 13164   Atomscatm 30075   HLchlt 30162   LHypclh 30795   LTrncltrn 30912
This theorem is referenced by:  ltrncnvat  30952
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-undef 6314  df-riota 6320  df-map 6790  df-plt 14108  df-glb 14125  df-p0 14161  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-hlat 30163  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916
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